Comparative Studies on Mesh-Free Deep Neural Network Approach Versus Finite Element Method for Solving Coupled Nonlinear Hyperbolic/Wave Equations
Year: 2022
Author: Xingwen Zhu, Mingyan He, Pengtao Sun
International Journal of Numerical Analysis and Modeling, Vol. 19 (2022), Iss. 5 : pp. 603–629
Abstract
In this paper, both the finite element method (FEM) and the mesh-free deep neural network (DNN) approach are studied in a comparative fashion for solving two types of coupled nonlinear hyperbolic/wave partial differential equations (PDEs) in a space of high dimension $\mathbb{R}^d (d > 1),$ where the first PDE system to be studied is the coupled nonlinear Korteweg-De Vries (KdV) equations modeling the solitary wave and waves on shallow water surfaces, and the second PDE system is the coupled nonlinear Klein-Gordon (KG) equations modeling solitons as well as solitary waves. A fully connected, feedforward, multi-layer, mesh-free DNN approach is developed for both coupled nonlinear PDEs by reformulating each PDE model as a least-squares (LS) problem based upon DNN-approximated solutions and then optimizing the LS problem using a $(d + 1)$-dimensional space-time sample point (training) set. Mathematically, both coupled nonlinear hyperbolic problems own significant differences in their respective PDE theories; numerically, they are approximated by virtue of a fully connected, feedforward DNN structure in a uniform fashion. As a contrast, a distinct and sophisticated FEM is developed for each coupled nonlinear hyperbolic system, respectively, by means of the Galerkin approximation in space and the finite difference scheme in time to account for different characteristics of each hyperbolic PDE system. Overall, comparing with the subtly developed, problem-dependent FEM, the proposed mesh-free DNN method can be uniformly developed for both coupled nonlinear hyperbolic systems with ease and without a need of mesh generation, though, the FEM can produce a concrete convergence order with respect to the mesh size and the time step size, and can even preserve the total energy for KG equations, whereas the DNN approach cannot show a definite convergence pattern in terms of parameters of the adopted DNN structure but only a universal approximation property indicated by a relatively small error that rarely changes in magnitude, let alone the dissipation of DNN-approximated energy for KG equations. Both approaches have their respective pros and cons, which are also validated in numerical experiments by comparing convergent accuracies of the developed FEMs and approximation performances of the proposed mesh-free DNN method for both hyperbolic/wave equations based upon different types of discretization parameters changing in doubling, and specifically, comparing discrete energies obtained from both approaches for KG equations.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2022-IJNAM-20929
International Journal of Numerical Analysis and Modeling, Vol. 19 (2022), Iss. 5 : pp. 603–629
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 27
Keywords: Coupled hyperbolic/wave equations Korteweg-De Vries (KdV) equations Klein-Gordon (KG) equations deep neural network (DNN) finite element method (FEM) space-time sample points (training) set least-squares (LS) convergence accuracy energy conservation.